A static site is one that you can serve through static hosting, where you have no control over the web server or its configuration. There is not some extra thing which is a static site with dynamic content. “Static” means “doesn't change.” The document served doesn't change subject to the person receiving it. You are talking about a solution that is dynamic. That does change based on who is making the request.

>you could definitely use a static site generator to create multiple versions of the site data and then configure your web server to select which data is emitted

And this web-server configuration would not exist within the static site. The static site generator could not output it, therefore it is not a part of the static site. It is not contained within the files output by the static site generator. It is additional dynamic content added by the web server.

It breaks the fundamental aspect of a static site, that it can be deployed simply to any service without change to the content. Just upload a zip file, and you are done.